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ACTSC445 Study Guide - Quiz Guide: Likelihood-Ratio Test, Independent And Identically Distributed Random Variables, Random VariableExam

Actuarial Science
Course Code
Jiahua Chen
Study Guide

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ACTSC 445/845 – Fall 2014
ASSIGNMENT 3 DUE:10:55 am Thursday November 27th
Hand in to Yunran Wei in the TA office.
You may work on this assignment individually or in groups of up to three people. If
you work in a group, you should submit a single solution with each students’ name
and ID number.
If you work in a group, each student in the group will receive the same mark.
There will be no extra credit for students who work individually.
For use in both questions:
Download a data set of weekly returns from for the period of 1990-2013 for two
stocks, A and B. Use financial related industries (banks, insurers, etc). See the note at the
end of the assignment for details of how to do this.
1. For stock A and stock B, separately:
We are interested in studying extreme outcomes of the negative logreturns, per
$100 invested, which is denoted by Xi=100 log(Si+1/Si) for the ith week, where
Siis your stock price. You may assume that the log-returns are iid with a common
distribution F.
Let Mj,j= 1,2, ..., n denote the maximum values of Xiin the jth block, where each
block is 26 weeks long.
(a) Generate a histogram of the observed Mj.
(b) Using maximum likelihood, estimate the GEV parameters for the distribution.
(c) Compare the data and the tted model with a Q-Q plot, and comment on the t.
(d) Use the likelihood ratio test to compare the simpler model, γ= 0 with the model
γ > 0, and comment on the results.
(e) Use sample mean excess plot to nd an appropriate threshold uso that the ex-
ceedances can be tted with a generalized Pareto distribution (GPD).
(f) Fit the GPD to the exceedances of the negative logreturns.
(g) Test the hypothesis γ= 0, using the likelihood ratio test.
(h) Based on the GPD you obtained in the previous steps, produce a mean excess
plot of the negative logreturns.
(i) Develop estimates for V aR0.99(X) and CTE0.99(X) by using the PGD you devel-
oped, where Xdenotes the negative logreturn random variable.
(j) Compare the results for Stock A and Stock B.
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